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A226133 Integers of the form (pq-1)/24 where p < q are primes. 1
6, 9, 11, 20, 21, 23, 27, 29, 30, 31, 33, 34, 36, 37, 38, 41, 44, 45, 49, 53, 56, 58, 59, 60, 61, 63, 64, 65, 66, 68, 79, 80, 81, 82, 85, 94, 96, 97, 98, 102, 104, 106, 107, 110, 115, 116, 120, 122, 124, 128, 129 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Results for p = q are given in A024702, which is complementary.

All integer results when viewed in the triangle occur in loosely diagonal, interrupted "bands" roughly (or exactly) parallel to main diagonal, such that  q - p = 24m, where m = 1 for the first band closest to the main diagonal,  m = 2 for the second band, m = 3 for the third band, etc.  The main diagonal p = q can be considered as fitting in this pattern where m = 0.

A general "rule" can be stated: If q-p = 24m for any m >= 0 and primes p < q, then pq-1 is divisible by 24.  This follows algebraically from the know "rule" that p^2 -1 is divisible by 24 for any prime p > 3 as given in A024702.

No result will occur twice, even when including A024702, because the product of any two primes is unique within the set.

Integer results have a density of about 12% to 13% for all possible p,q pairs among the first few hundred primes.

LINKS

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

EXAMPLE

(5*29-1)/24 = 6,  (7*31-1)/24 = 9,  (5*53-1)/24 = 11,  also note about these three examples, in order, that 29-5 = 24, 31-7 = 24 and 53-5 = 48.

PROG

(PARI) is(n)=my(f=factor(24*n+1)); #f[, 1]==2&&f[1, 2]==1&&f[2, 2]==1 \\ Charles R Greathouse IV, May 30 2013

CROSSREFS

Complementary to A024702.

Sequence in context: A275274 A131834 A084418 * A125971 A322158 A143710

Adjacent sequences:  A226130 A226131 A226132 * A226134 A226135 A226136

KEYWORD

nonn

AUTHOR

Richard R. Forberg, May 27 2013

EXTENSIONS

Missing a(8) from Charles R Greathouse IV, May 31 2013

STATUS

approved

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Last modified March 20 13:18 EDT 2019. Contains 321345 sequences. (Running on oeis4.)